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Let $M$ be a finitely generated torsion-free module over a generalized Dedekind domain $D$. It is shown that if $M$ is a projective $D$-module, then it is a generalized Dedekind module and $v$-multiplication module. In case $D$ is Noetherian it is shown that $M$ is either a generalized Dedekind module or a Krull module. Furthermore, the polynomial module $M[x]$ is a generalized Dedekind $D[x]$-module (a Krull $D[x]$-module) if $M$ is a generalized Dedekind module (a Krull module), respectively.

Let $M$ be a torsion-free module over an integral domain $D$ with quotient field $K$. We define a concept of completely integrally closed modules in order to study Krull modules. It is shown that a Krull module $M$ is a $v$-multiplication module if and only if ${(𝔭M)}_v$ is a maximal $v$-submodule and ${({𝔭}^{-1}𝔭)}_{v_1}=D$ for every minimal prime ideal $𝔭$ of $D$. If $M$ is a finitely generated Krull module, then $M_1=\cap M_{𝔭}$ is a Krull module and $v$-multiplication module. It is also shown that the following three conditions are equivalent: $M$ is completely integrally closed, $M[x]$ is completely integrally closed, and $M[[x]]$ is completely integrally closed.

We define a concept of strongly Krull (s-Krull) modules over integral domains. One of the main aims is to study arithmetic properties of s-Krull modules. There are two different definitions of Krull modules over integral domains, that is: Krull modules in the sense of Costa and Johnson and Krull modules in the sense of ours. It turns out that Krull modules in the sense of Costa and Johnson imply s-Krull modules and s-Krull modules imply Krull modules in the sense of ours. The reverse of the statements do not necessarily hold.

In this article, we generalize the concepts of several classes of domains (which are related to Krull domains) to torsion-free modules, and show that for a faithful multiplication module M over an integral domain R, M is a Krull module if and only if R is a Krull domain. Then we characterize Krull, Dedekind, and factorial modules.